Download Lesson 2-5: Reasoning in Algebra and Geometry Goal

LESSON 2-5: REASONING IN
ALGEBRA AND GEOMETRY
Goal: To connect reasoning in Algebra and
Geometry.
PROPERTIES OF EQUALITY:
Let 𝑎, 𝑏, and 𝑐 be any real numbers.
Addition Property
Subtraction Property
Multiplication Property
Division Property
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐.
𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 = 𝑏 − 𝑐.
𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 ∙ 𝑐 = 𝑏 ∙ 𝑐.
𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛
𝑏
= 𝑐.
𝑎=𝑎
𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎.
𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.
𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 𝑐𝑎𝑛 𝑟𝑒𝑝𝑙𝑎𝑐𝑒 𝑎
𝑖𝑛 𝑎𝑛𝑦 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛.
Distributive Property
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐
𝑎
𝑐
𝑎 𝑏 − 𝑐 = 𝑎𝑏 − 𝑎𝑐
EXAMPLE 1: JUSTIFYING STEPS WHEN
SOLVING AN EQUATION

What is the Value of x? Justify each step.
∠𝐴𝑂𝑀 𝑎𝑛𝑑 ∠𝑀𝑂𝐶 are supplementary ∠’s that form a linear pair
are supplementary
Definition of supplementary ∠′ 𝑠
𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180
Substitution property
2𝑥 + 30 + 𝑥 = 180
Combining like terms
3𝑥 + 30 = 180
Subtraction Property of Equality
3𝑥 = 150
Division Property of Equality
𝑥 = 50
PROPERTIES OF CONGRUENCE:
Reflexive Property
𝐴𝐵 ≅ 𝐴𝐵
Symmetric Property
𝐼𝑓 𝐴𝐵 ≅ 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 ≅ 𝐴𝐵
𝐼𝑓 ∠𝐴 ≅ ∠𝐵, 𝑡ℎ𝑒𝑛 ∠𝐵 ≅ ∠𝐴
Transitive Property
𝐼𝑓 𝐴𝐵 ≅ 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 ≅ 𝐸𝐹,
𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐸𝐹
𝐼𝑓 ∠𝐴 ≅ ∠𝐵 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐶,
𝑡ℎ𝑒𝑛 ∠𝐴 ≅ ∠𝐶.
∠𝐴 ≅ ∠𝐴
EXAMPLE 2: USING PROPERTIES OF
EQUALITY AND CONGRUENCE

What is the property of congruence that justifies
going from the first statement to the second
statement?

2𝑥 + 9 = 19
2𝑥 = 10

∠𝑂 ≅ ∠𝑊 𝑎𝑛𝑑 ∠𝑊 ≅ ∠𝐿
∠𝑂 ≅ ∠𝐿
Transitive Prop. Of Congruence

𝑚∠𝐸 = 𝑚∠𝑇
𝑚∠𝑇 = 𝑚∠𝐸
Symmetric Prop. Of Equality
Subtraction Prop. Of Equality
EXAMPLE 3:
WRITING A TWO-COLUMN PROOF
ASSIGNMENT:

Pg 117 – 118 #’s 6-12 even, 14-19 all and 25